BOUNDED LINEAR OPERATORS THAT COMMUTE WITH SHIFTS ARE SCALED IDENTITY

We prove that every bounded linear operator on L^p?L^p(R'),s≥1 and 1≤p<∞, that commutes with both the spatial shift operator and the phase shift operator must be a constant multiple of the identity. We also apply this result to identify analysis-synthesis dual L^q-L^p paris and to characterize bi-orthogonal unconditional bases of L^q-L^p, where p^?1+q^?1=1.     

β-delayed proton decay of89Ru

⁸⁹Ru was synthesized in the reaction of⁵⁸Ni (³⁶Ar, 2p3n) on the basis of a “p-y” coincidence measurement by using a He-jet tape transport system, and its β-delayed proton emission with a half-life of (1.1 ± 0.2) s was investigated. The β-delayed proton spectrum of⁸⁹Ru populating the lowlying states in⁸⁸Mo was obtained, and the final state proton branching ratios to the low-lying 2⁺ and 4⁺ states in⁸⁸Mo were estimated to be 100:6. Based on the statistical model calculations, the ground state spin of⁸⁹Ru was preliminarily assigned to be 5/2⁺ or 7/2^±, and the mass excess of⁸⁹Ru was deduced to be -59.5 MeV.     

LEGENDRE TRANSFORMATIONS AND CARTAN FORMS IN THE CALCULUS OF VARIATIONS OF MULTIPLE INTEGRALS

ABSTRACT

(PART II): In terms of a given Hamiltonian function the 1-form w = dH + ?_j|dπ^j is defined, where {?_j:j = 1,…, n} denotes an invariant basis of the planes of the distribution D_n. The latter is said to be canonical if w = 0 (which is analogous to the definition of Hamiltonian vector fields in symplectic geometry). This condition is equivalent to two sets of canonical equations that are expressed explicitly in term of the derivatives of H with respect to its positional arguments. The distribution D_n is said to be pseudo-Lagrangian if dπ^j(?_j,V_h) = 0; if D_n, is both canonical and pseudo-Lagrangian it is integrable and such that H = const. on each leaf of the resulting foliation. The Cartan form associated with this construction [9] is defined a II = π² ? ? πⁿ. If π is closed, the distribution D_N is integrable, and the exterior system {π^j} admits the representation ψ^j = dS^j in terms of a set of 0-forms S^j on M. If, in addition, the distribution D_N is canonical, these functions satisfy a single first order Hamilton-Jacobi equation, and conversely. Finally, a complete figure is constructed on the basis of the assumptions that (i) the Cartan form be closed, and (ii) that the distribution D_n, be both canonical and integrable. The last of these requirements implies the existence of N functions ψ^A that depend on x^h and N parameters w^B, whose derivatives are given by ?ψ^A (x^h, w^B)/?x^j = B^A _j (x^h, ψ^B (x^h,w^B)). The complete figure then consists of two complementary foliations: the leaves of the first are described by the functions ψ^A and satisfy the standard Euler-Lagrange equations, while the second, that is, the transversal foliation, is represented by the aforementioned solution of the Hamilton-Jacobi equation. The entire configuration then gives rise in a natural manner to a generalized Hilbert independent integral and consequently also to a generalized Weierstrass excess function.     

β-delayed proton decay of89Ru

⁸⁹Ru was synthesized in the reaction of⁵⁸Ni (³⁶Ar, 2p3n) on the basis of a “p-y” coincidence measurement by using a He-jet tape transport system, and its β-delayed proton emission with a half-life of (1.1 ± 0.2) s was investigated. The β-delayed proton spectrum of⁸⁹Ru populating the lowlying states in⁸⁸Mo was obtained, and the final state proton branching ratios to the low-lying 2⁺ and 4⁺ states in⁸⁸Mo were estimated to be 100:6. Based on the statistical model calculations, the ground state spin of⁸⁹Ru was preliminarily assigned to be 5/2⁺ or 7/2^±, and the mass excess of⁸⁹Ru was deduced to be -59.5 MeV.     

A Transportation model with multiple criteria and multiple constraint levels

In this paper, a transportation model with multiple criteria and multiple constraint levels (MC²) is formulated by using the framework of MC² linear programming. An algorithm is developed to solve such MC² transportation problems. In this algorithm, the traditional northwest corner rule is adopted to find an initial basic feasible solution for a given MC² transportation problem. Then the MC²-simplex method is applied to locate the set of all potential solutions over possible changes of the objective coefficient parameter and the supply and demand parameter for the MC² transportation problem. A numerical example is illustrated to demonstrate the applicability of the algorithm in solving the MC² transportation problems.     

Three-dimensional hyperelliptic manifolds

Let X³ = H³, E³, S³, H² × E¹, S² × E¹, T₁(H²), Nil of Solv be one of the eight 3-dimensional geometrics of Thurston [10] and G be a discrete group of isometrics of X³ acting without fixed points. A manifold M³ = X³/G is said to be hyperelliptic if there is an isometric involution on it such that the factor space M³/<> is diffeomorphic to the 3-sphere S³. In analogy with the theory of Riemann surfaces we call involution.In the present paper the existence of hyperelliptic manifolds in each light of the eight 3-dimensional geometrics will be obtained. All the proofs given there will be written in the language of orbifolds whose basic facts can be found in [9].     

New β-delayed proton decays in the rare-earth region near the proton drip line

The β-delayed proton precursors 125Nd, 128Pm, 129Sm, 137Gd and 139Dy near the proton drip line were produced by the irradiation of 92Mo, 96Ru and 106Cd with an 36Ar beam, and conclusively identified for the first time by using proton-gamma coincidence in combination with a He-jet tape transport system. Their half-lives were determined to be 0.60(15)s,1.0(3)s,0.55(10)s,2.2(2)s and 0.6(2)s, respectively. The measured energy spectra of β-delayed protons and estimated proton branching ratios to the final states in "daughter" nuclei for the precursors 125Nd, 129Sm, 137Gd and 139Dy, as well as the previously reported ones for the precursors 135Gd and 121Ce, were fitted by a statistical model calculation. The ground-state spins and parities of 121Ce, 125Nd, 129Sm, 135Gd, 137Gd and 139Dy were then assigned as 5/2±,5/2±,1/2±(or 3/2+),5/2+,7/2± and 7/2, respectively. The consistency between the experimental spin-parity assignment and the predicted Nilsson diagrams indirectly indicates that the ground states of 121Ce, 125Nd, 129Sm, 135Gd, 137Gd and 139Dy are highly deformed with β2～0.3.     

Separability and entanglement in tripartite states

While classical correlations can be freely distributed among many systems, this is not true for entanglement and quantum correlations. If a quantum system S^a is entangled with another quantum system S^b, then its entanglement with any third quantum system S^c cannot be arbitrary. This is the celebrated monogamy of entanglement. Implicit in this general statement is the plausible belief that only entanglement between the systems S^a and S^b constrains the entanglement between S^a and the third system S^c. We demonstrate that even classical correlations between S^a and S^b may impose surprisingly stringent restrictions on the possible entanglement between S^a and S^c. In particular, perfect bipartite classical correlations and full entanglement cannot coexist in any tripartite state. An intuitive explanation of this monogamy of hybrid classical and quantum correlations might be that the system S^a has a correlating capability, which cannot be used to establish any entanglement with a third system (but can still be used to establish classical correlations) if it is exhausted when correlated with S^b (in either a classical or quantum fashion). This may be interpreted as an alternate version of monogamy.     

Final state interactions in the decay of light hypernuclei

The excited states of Li⁴, Li⁵ and Be⁸ have been investigated by studying the final state interactions in the π^?-mesic decays of_ΛHe⁴,_ΛHe⁵ and_ΛLi⁸ respectively. The experimental results indicate the existence of final states like Li^5* at 1·8 MeV and Be^8* at 0 and 2·9 MeV to be dominant in the decay of_ΛHe^5* and_ΛLi⁸ respectively; the results on the decay of_ΛHe⁴ suggest the presence of a Li^4* state at about 4 MeV to dominate.     

Conjugate function theory in weak1 Dirichlet algebras

The fundamental theorems on conjugate functions are shown to be valid for weak¹ Dirichlet algebras. In particular the conjugation operator is shown to be a continuous map of L^p to L^p for 1 < p < ∞, to be a continuous map of L¹ to L^p, 0 < p < 1, and to map functions in L^∞ to exponentially integrable functions. These results allow a number of results for Dirichlet algebras to be extended to weak¹ Dirichlet algebras.     

New class of polynomials

Summary For the simple Bessel polynomials y_n(x), the Rodrigues' formula is y_n(x)==2⁻ⁿe^2/xDⁿ(x²ⁿe^−2/x). The present work generalizes the idea of the simple. Bessel polynomials of Krall-Frink. It is of great interest to investigate the class of polynomials which are closely connected with the Rodrigues' expression e^3/xDⁿ(x³ⁿe^−3/x) or more generally with e^k/xDⁿ(x^kne^−k/x).     

Characterization ofL p (R n ) using Gabor frames

We characterize L^p norms of functions onR ⁿ for 1<p<∞ in terms of their Gabor coefficients. Moreover, we use the Carleson-Hunt theorem to show that the Gabor expansions of L^p functions converge to the functions almost everywhere and in L^p for 1<p<∞. In L¹ we prove an analogous result: the Gabor expansions converge to the functions almost everywhere and in L¹ in a certain Cesàro sense. Consequently, we are able to establish that a large class of Gabor families generate Banach frames for L^p (R ⁿ) when 1≤p<∞.     

On Integrable Delta Functions on the Levi-Civita Field

In this paper, we develop a theory of integrable delta functions on the Levi-Civita field R as well as on R² and R³ with similar properties to the one-dimensional, two-dimensional and three-dimensional Dirac Delta functions and which reduce to them when restricted to points in R, R² and R³, respectively. First we review the recently developed Lebesgue-like measure and integration theory over R, R² and R³. Then we introduce delta functions on R, R² and R³ that are integrable in the context of the aforementioned integration theory; and we study their properties and some applications.     

Hodge Numbers of a Hypothetical Complex Structure on the Six Sphere

We prove that the terms E_r ^p,q(S⁶) in the Frölicher spectral sequence associated to any hypothetical complex structure on S⁶ would satisfy Serre duality. It is also shown that the vanishing of the Dolbeault cohomology group H^1,1(S⁶) ensures the existence of a holomorphic 2-form on S⁶ living even in E₂ ^2,0(S⁶), which in particular implies the nondegeneration of Frölicher's sequence at the second level.     

A Deodhar-type stratification on the double flag variety

We describe a partition of the double flag variety G/B⁺ × G/B^- of a complex semisimple algebraic group G analogous to the Deodhar partition on the flag variety G/B⁺. This partition is a refinement of the stratification into orbits both for B⁺ × B^- and for the diagonal action of G, just as Deodhar's partition refines the orbits of B⁺ and B^-. We give a coordinate system on each stratum, and show that all strata are coisotropic subvarieties. Also, we discuss possible connections to the positive and cluster geometry of G/B⁺ × G/B^-, which would generalize results of Fomin and Zelevinsky on double Bruhat cells and Marsh and Rietsch on double Schubert cells.     

Internal conversion coefficients of E2 transitions

An analysis of K-conversion coefficients of E2 transitions of the type 2⁺→0⁺, 0⁺→2⁺, 4⁺→2⁺, 6⁺→4⁺, and 8⁺→6⁺ together with their methods of determination has been carried out. The ratios of experimental to theoreticalα _k values are plotted against the atomic weights and it is seen that there is no similar behaviour among these transitions.     

Determination of the flux of primary cosmic ray hydrogen and helium nuclei near the geomagnetic equator using nuclear emulsions

Nuclear emulsions exposed to the cosmic radiation over Hyderabad, India, at an altitude of 31·3 km. for six hours, have been used to derermine primary cosmic ray deuteron flux. The flux of deuterons of rigidity ?16·9 GV at the top of the atmosphere is found to be <10 m.^?2 sec.^?1 sterad^?1; this may be compared with the proton fiux of 83±12 m.^?2 sec.^?1 sterad.^?1 in the same rigidity region over Hyderabad.     

A new upper bound on the cyclic chromatic number

A cyclic coloring of a plane graph is a vertex coloring such that vertices incident with the same face have distinct colors. The minimum number of colors in a cyclic coloring of a graph is its cyclic chromatic number χ^c. Let Δ^* be the maximum face degree of a graph. There exist plane graphs with χ^c = ?3/2 Δ^*?. Ore and Plummer [ 5 ] proved that χ^c ≤ 2, Δ^*, which bound was improved to ?9/5, Δ^*? by Borodin, Sanders, and Zhao [ 1 ], and to ?5/3,Δ^*? by Sanders and Zhao [ 7 ]. We introduce a new parameter k^*, which is the maximum number of vertices that two faces of a graph can have in common, and prove that χ^c ≤ max {Δ^* + 3,k^* + 2, Δ^* + 14, 3, k^* + 6, 18}, and if Δ^* ≥ 4 and k^* ≥ 4, then χ^c ≤ Δ^* + 3,k^* + 2. © 2006 Wiley Periodicals, Inc. J Graph Theory     

Large time behavior of solutions for critical and subcritical complex Ginzburg-Landau equations in H<Superscript>1</Superscript>

Considering the Cauchy problem for the critical complex Ginzburg-Landau equation in H¹(Rⁿ), we shall show the asymptotic behavior for its solutions in C(0, ∖;H¹(Rⁿ)) ∩ L²(0, ∖;H^1,2n/(n-2)(R²)), n≥3. Analogous results also hold in the case that the nonlinearity has the subcritical power in H¹(Rⁿ), n≥1. Dedicated to Professor Zhou Yulin for his 80th birthday.